setfunc/sps_log.hpp

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• Last update: 2023-11-21 19:08:32+09:00
• Include: #include "setfunc/sps_log.hpp"

Depends on

• mod/modint.hpp
• mod/modint_common.hpp
• setfunc/ranked_zeta.hpp
• setfunc/sps_composition.hpp

Verified with

• test_atcoder/abc213g.test.cpp
• test_atcoder/abc321g.test.cpp
• test_atcoder/arc105.test.cpp

Code

#include "mod/modint.hpp"
#include "setfunc/sps_composition.hpp"

// exp の逆手順で計算する
template <typename mint, int LIM>
vc<mint> sps_log(vc<mint>& dp) {
  const int N = topbit(len(dp));
  assert(len(dp) == (1 << N) && dp[0] == mint(1));
  vc<mint> s(1 << N);
  FOR_R(i, N) {
    vc<mint> a = {dp.begin() + (1 << i), dp.begin() + (2 << i)};
    vc<mint> b = {dp.begin(), dp.begin() + (1 << i)};
    auto RA = ranked_zeta<mint, LIM>(a);
    auto RB = ranked_zeta<mint, LIM>(b);
    FOR(s, 1 << i) {
      auto &f = RA[s], &g = RB[s];
      // assert(g[0] == mint(1));
      FOR(d, i + 1) { FOR(i, d) f[d] -= f[i] * g[d - i]; }
    }
    a = ranked_mobius<mint, LIM>(RA);
    copy(all(a), s.begin() + (1 << i));
  }
  return s;
}

#line 2 "mod/modint_common.hpp"

struct has_mod_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{});
  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_mod : public decltype(has_mod_impl::check<T>(std::declval<T>())) {};

template <typename mint>
mint inv(int n) {
  static const int mod = mint::get_mod();
  static vector<mint> dat = {0, 1};
  assert(0 <= n);
  if (n >= mod) n %= mod;
  while (len(dat) <= n) {
    int k = len(dat);
    int q = (mod + k - 1) / k;
    dat.eb(dat[k * q - mod] * mint::raw(q));
  }
  return dat[n];
}

template <typename mint>
mint fact(int n) {
  static const int mod = mint::get_mod();
  assert(0 <= n && n < mod);
  static vector<mint> dat = {1, 1};
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat)));
  return dat[n];
}

template <typename mint>
mint fact_inv(int n) {
  static vector<mint> dat = {1, 1};
  if (n < 0) return mint(0);
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv<mint>(len(dat)));
  return dat[n];
}

template <class mint, class... Ts>
mint fact_invs(Ts... xs) {
  return (mint(1) * ... * fact_inv<mint>(xs));
}

template <typename mint, class Head, class... Tail>
mint multinomial(Head &&head, Tail &&... tail) {
  return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...);
}

template <typename mint>
mint C_dense(int n, int k) {
  static vvc<mint> C;
  static int H = 0, W = 0;
  auto calc = [&](int i, int j) -> mint {
    if (i == 0) return (j == 0 ? mint(1) : mint(0));
    return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0);
  };
  if (W <= k) {
    FOR(i, H) {
      C[i].resize(k + 1);
      FOR(j, W, k + 1) { C[i][j] = calc(i, j); }
    }
    W = k + 1;
  }
  if (H <= n) {
    C.resize(n + 1);
    FOR(i, H, n + 1) {
      C[i].resize(W);
      FOR(j, W) { C[i][j] = calc(i, j); }
    }
    H = n + 1;
  }
  return C[n][k];
}

template <typename mint, bool large = false, bool dense = false>
mint C(ll n, ll k) {
  assert(n >= 0);
  if (k < 0 || n < k) return 0;
  if constexpr (dense) return C_dense<mint>(n, k);
  if constexpr (!large) return multinomial<mint>(n, k, n - k);
  k = min(k, n - k);
  mint x(1);
  FOR(i, k) x *= mint(n - i);
  return x * fact_inv<mint>(k);
}

template <typename mint, bool large = false>
mint C_inv(ll n, ll k) {
  assert(n >= 0);
  assert(0 <= k && k <= n);
  if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k);
  return mint(1) / C<mint, 1>(n, k);
}

// [x^d](1-x)^{-n}
template <typename mint, bool large = false, bool dense = false>
mint C_negative(ll n, ll d) {
  assert(n >= 0);
  if (d < 0) return mint(0);
  if (n == 0) { return (d == 0 ? mint(1) : mint(0)); }
  return C<mint, large, dense>(n + d - 1, d);
}
#line 3 "mod/modint.hpp"

template <int mod>
struct modint {
  static constexpr u32 umod = u32(mod);
  static_assert(umod < u32(1) << 31);
  u32 val;

  static modint raw(u32 v) {
    modint x;
    x.val = v;
    return x;
  }
  constexpr modint() : val(0) {}
  constexpr modint(u32 x) : val(x % umod) {}
  constexpr modint(u64 x) : val(x % umod) {}
  constexpr modint(u128 x) : val(x % umod) {}
  constexpr modint(int x) : val((x %= mod) < 0 ? x + mod : x){};
  constexpr modint(ll x) : val((x %= mod) < 0 ? x + mod : x){};
  constexpr modint(i128 x) : val((x %= mod) < 0 ? x + mod : x){};
  bool operator<(const modint &other) const { return val < other.val; }
  modint &operator+=(const modint &p) {
    if ((val += p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator-=(const modint &p) {
    if ((val += umod - p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator*=(const modint &p) {
    val = u64(val) * p.val % umod;
    return *this;
  }
  modint &operator/=(const modint &p) {
    *this *= p.inverse();
    return *this;
  }
  modint operator-() const { return modint::raw(val ? mod - val : u32(0)); }
  modint operator+(const modint &p) const { return modint(*this) += p; }
  modint operator-(const modint &p) const { return modint(*this) -= p; }
  modint operator*(const modint &p) const { return modint(*this) *= p; }
  modint operator/(const modint &p) const { return modint(*this) /= p; }
  bool operator==(const modint &p) const { return val == p.val; }
  bool operator!=(const modint &p) const { return val != p.val; }
  modint inverse() const {
    int a = val, b = mod, u = 1, v = 0, t;
    while (b > 0) {
      t = a / b;
      swap(a -= t * b, b), swap(u -= t * v, v);
    }
    return modint(u);
  }
  modint pow(ll n) const {
    assert(n >= 0);
    modint ret(1), mul(val);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
  static constexpr int get_mod() { return mod; }
  // (n, r), r は 1 の 2^n 乗根
  static constexpr pair<int, int> ntt_info() {
    if (mod == 120586241) return {20, 74066978};
    if (mod == 167772161) return {25, 17};
    if (mod == 469762049) return {26, 30};
    if (mod == 754974721) return {24, 362};
    if (mod == 880803841) return {23, 211};
    if (mod == 943718401) return {22, 663003469};
    if (mod == 998244353) return {23, 31};
    if (mod == 1045430273) return {20, 363};
    if (mod == 1051721729) return {20, 330};
    if (mod == 1053818881) return {20, 2789};
    return {-1, -1};
  }
  static constexpr bool can_ntt() { return ntt_info().fi != -1; }
};

#ifdef FASTIO
template <int mod>
void rd(modint<mod> &x) {
  fastio::rd(x.val);
  x.val %= mod;
  // assert(0 <= x.val && x.val < mod);
}
template <int mod>
void wt(modint<mod> x) {
  fastio::wt(x.val);
}
#endif

using modint107 = modint<1000000007>;
using modint998 = modint<998244353>;
#line 2 "setfunc/ranked_zeta.hpp"

template <typename T, int LIM>
vc<array<T, LIM + 1>> ranked_zeta(const vc<T>& f) {
  int n = topbit(len(f));
  assert(n <= LIM);
  assert(len(f) == 1 << n);
  vc<array<T, LIM + 1>> Rf(1 << n);
  for (int s = 0; s < (1 << n); ++s) Rf[s][popcnt(s)] = f[s];
  for (int i = 0; i < n; ++i) {
    int w = 1 << i;
    for (int p = 0; p < (1 << n); p += 2 * w) {
      for (int s = p; s < p + w; ++s) {
        int t = s | 1 << i;
        for (int d = 0; d <= n; ++d) Rf[t][d] += Rf[s][d];
      }
    }
  }
  return Rf;
}

template <typename T, int LIM>
vc<T> ranked_mobius(vc<array<T, LIM + 1>>& Rf) {
  int n = topbit(len(Rf));
  assert(len(Rf) == 1 << n);
  for (int i = 0; i < n; ++i) {
    int w = 1 << i;
    for (int p = 0; p < (1 << n); p += 2 * w) {
      for (int s = p; s < p + w; ++s) {
        int t = s | 1 << i;
        for (int d = 0; d <= n; ++d) Rf[t][d] -= Rf[s][d];
      }
    }
  }
  vc<T> f(1 << n);
  for (int s = 0; s < (1 << n); ++s) f[s] = Rf[s][popcnt(s)];
  return f;
}
#line 2 "setfunc/sps_composition.hpp"

// sum_i f_i/i! s^i, s^i is subset-convolution
template <typename mint, int LIM>
vc<mint> sps_composition_egf(vc<mint>& f, vc<mint>& s) {
  const int N = topbit(len(s));
  assert(len(s) == (1 << N) && s[0] == mint(0));
  if (len(f) > N) f.resize(N + 1);
  int D = len(f) - 1;
  using ARR = array<mint, LIM + 1>;
  vvc<ARR> zs(N);
  FOR(i, N) {
    zs[i]
        = ranked_zeta<mint, LIM>({s.begin() + (1 << i), s.begin() + (2 << i)});
  }

  // dp : (d/dt)^df(s) (d=D,D-1,...)
  vc<mint> dp(1 << (N - D));
  dp[0] = f[D];
  FOR_R(d, D) {
    vc<mint> newdp(1 << (N - d));
    newdp[0] = f[d];
    vc<ARR> zdp = ranked_zeta<mint, LIM>(dp);
    FOR(i, N - d) {
      // zs[1<<i:2<<i], zdp[0:1<<i]
      vc<ARR> znewdp(1 << i);
      FOR(k, 1 << i) {
        FOR(p, i + 1) FOR(q, i - p + 1) {
          znewdp[k][p + q] += zdp[k][p] * zs[i][k][q];
        }
      }
      auto x = ranked_mobius<mint, LIM>(znewdp);
      copy(all(x), newdp.begin() + (1 << i));
    }
    swap(dp, newdp);
  }
  return dp;
}

// sum_i f_i s^i, s^i is subset-convolution
template <typename mint, int LIM>
vc<mint> sps_composition_poly(vc<mint> f, vc<mint> s) {
  const int N = topbit(len(s));
  assert(len(s) == (1 << N));
  if (f.empty()) return vc<mint>(1 << N, mint(0));
  // convert to egf problem
  int D = min<int>(len(f) - 1, N);
  vc<mint> g(D + 1);
  mint c = s[0];
  s[0] = 0;
  // (x+c)^i
  vc<mint> pow(D + 1);
  pow[0] = 1;
  FOR(i, len(f)) {
    FOR(j, D + 1) g[j] += f[i] * pow[j];
    FOR_R(j, D + 1) pow[j] = pow[j] * c + (j == 0 ? mint(0) : pow[j - 1]);
  }
  // to egf
  mint factorial = 1;
  FOR(j, D + 1) g[j] *= factorial, factorial *= mint(j + 1);
  return sps_composition_egf<mint, LIM>(g, s);
}
#line 3 "setfunc/sps_log.hpp"

// exp の逆手順で計算する
template <typename mint, int LIM>
vc<mint> sps_log(vc<mint>& dp) {
  const int N = topbit(len(dp));
  assert(len(dp) == (1 << N) && dp[0] == mint(1));
  vc<mint> s(1 << N);
  FOR_R(i, N) {
    vc<mint> a = {dp.begin() + (1 << i), dp.begin() + (2 << i)};
    vc<mint> b = {dp.begin(), dp.begin() + (1 << i)};
    auto RA = ranked_zeta<mint, LIM>(a);
    auto RB = ranked_zeta<mint, LIM>(b);
    FOR(s, 1 << i) {
      auto &f = RA[s], &g = RB[s];
      // assert(g[0] == mint(1));
      FOR(d, i + 1) { FOR(i, d) f[d] -= f[i] * g[d - i]; }
    }
    a = ranked_mobius<mint, LIM>(RA);
    copy(all(a), s.begin() + (1 << i));
  }
  return s;
}

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